A wheel with mass m = 3 kg, radius R = 0.3 m rolls flat surface under the influence of a force F = 2 N applied to the center of the wheel and directed parallel to the plane. Initial speed V0=2m/s.  Rolling friction coefficient K1=0,018.  Slipping friction coefficient K2=0.018.  Under what conditions does wheel slip occur?

To determine the conditions under which the wheel begins to slip, we need to compare the force responsible for rolling the wheel (which includes the force \( F \) applied to the center of the wheel) and the force resisting the motion, particularly the rolling and slipping friction forces.

### Given Data:
- Mass of the wheel: \( m = 3 \, \text{kg} \)
- Radius of the wheel: \( R = 0.3 \, \text{m} \)
- Force applied at the center of the wheel: \( F = 2 \, \text{N} \)
- Initial speed: \( V_0 = 2 \, \text{m/s} \)
- Rolling friction coefficient: \( K_1 = 0.018 \)
- Slipping friction coefficient: \( K_2 = 0.018 \)

#### Step 1: Understanding Rolling and Slipping Friction
- **Rolling friction** is the force that resists the rolling motion and is proportional to the normal force \( F_N \). It is given by:
  \[
  F_{\text{rolling friction}} = K_1 F_N
  \]
  where \( F_N = mg \), the normal force, and \( g \) is the acceleration due to gravity.
  \[
  F_{\text{rolling friction}} = K_1 m g
  \]

- **Slipping friction** acts if the wheel starts slipping instead of purely rolling. It is given by:
  \[
  F_{\text{slipping friction}} = K_2 F_N = K_2 m g
  \]
  This force defines the threshold beyond which slipping will occur. As long as the frictional force needed to maintain rolling does not exceed this value, pure rolling occurs.

#### Step 2: Forces Acting on the Wheel
The force \( F \) is attempting to accelerate the wheel both translationally and rotationally. The total resistive force consists of rolling friction and any rotational inertia the wheel might have.

We need to check if the applied force \( F \) overcomes the resistive force provided by rolling friction and if the force on the point of contact exceeds the slipping friction.

#### Step 3: Slipping Condition
The wheel will begin to slip when the torque generated by the friction force is insufficient to maintain the no-slip condition, and this occurs when:
\[
F_{\text{applied}} > F_{\text{slipping friction}}
\]
Substituting the given values into this equation:
\[
F_{\text{slipping friction}} = K_2 m g = 0.018 \times 3 \times 9.8 = 0.5292 \, \text{N}
\]
Thus, the force needed to induce slipping must exceed 0.5292 N.

### Condition for Slipping
The wheel will begin to slip if the applied force \( F = 2 \, \text{N} \) exceeds the slipping friction threshold \( F_{\text{slipping friction}} = 0.5292 \, \text{N} \).

Since \( F = 2 \, \text{N} \) is greater than \( 0.5292 \, \text{N} \), **the wheel will slip** under these conditions. 

#### Final Condition:
The wheel will slip if the force applied to the center of the wheel is greater than the slipping friction force, which in this case is true. So, slipping occurs with the given values.

Would you like more detailed steps on frictional forces or rotational dynamics? Let me know if you have any questions!

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### Further Related Questions:
1. How is rolling friction different from slipping friction in terms of its effect on motion?
2. How do we calculate the moment of inertia for different types of wheels (solid disk vs. hollow cylinder)?
3. What happens to the angular velocity of the wheel when slipping starts?
4. How does the rolling friction coefficient affect the deceleration of the wheel?
5. Can a wheel with a higher mass slip more easily than a lighter one given the same force?

**Tip:** Always compare the applied force to both rolling and slipping friction to determine if a wheel will roll or slip!

A wheel with mass m = 3 kg, radius R = 0.3 m rolls on a flat surface under the influence of a force F = 2 N applied to the center of the wheel and directed parallel to the plane. Initial speed V0=2m/s. Rolling friction coefficient K1=0.018. Slipping friction coefficient K2=0.018. Under what conditions does the wheel slip occur?

Determine the conditions for a wheel with mass 3 kg and radius 0.3 m rolling on a flat surface under a force of 2 N to slip. This problem involves rolling and slipping friction, applying Newton's laws and rotational dynamics. Learn how to calculate the friction forces and predict when slipping occurs based on friction coefficients.

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